In \(n+m\) independent  Bernoulli \((p)\) trials, let \(S_n\) be the number of successes in the first \(n\) trials, \(T_n\) the number of successes in the last \(m\) trials.

1. What is the distribution of \(S_n\) ? Why ?
2. What is the distribution of \(T_m\) ? Why ?
3. What is the distribution of \(S_n+T_m\) ? Why ?
4. Are \(S_n\) and \(T_m\) independent ? Why ?
5. Are \(S_n\) and \(T_{m+1}\) independent ? Why ?
6. Are \(S_{n+1}\) and \(T_{m}\) independent ? Why ?

Based on [Pitman, p. 159, #10]